The key point of the proof of [1]} is the observation that
the orthogonal group \(O(H^{1,1}_{{Z}}(M), q)\)  of the lattice
\(H^{1,1}_{{Z}}(M)=H^{1,1}(M)\cap H^2(M,{Z})\) , and therefore the
Hodge monodromy group \(\Gamma _{\operatorname{\sf Hdg}}\) 
(see REF )
which is a subgroup of finite index in \(O(H^{1,1}_{{Z}}(M), q)\)  ,
acts with finitely many orbits on the set of classes of fixed square \(r\ne 0\) .
When the primitive MBM classes have bounded square, we conclude that the
monodromy
acts with finitely many orbits on the set of MBM classes. As those are
precisely the classes whose orthogonal hyperplanes support the faces
of the Kähler cone, it is not difficult to deduce that there are only
finitely many, up to the action of the monodromy group, faces of the Kähler cone,
and also finitely many oriented faces of the Kähler cone (an oriented face
is a face together with the choice of normal direction). An element of the
monodromy which sends a face \(F\)  to a face \(F^{\prime }\) , with both orientations pointing
towards the
interior of the Kähler cone, must preserve the Kähler cone.
On the other hand, Markman proved ([2]}, Theorem 1.3)
that an element of the Hodge monodromy which
preserves the Kähler cone must be induced by an automorphism, so that the cone
conjecture follows.
